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arXiv · 1411.4153

Petviashvilli's Method for the Dirichlet Problem

Derek Olson·Soumitra Shukla·Gideon Simpson·Daniel Spirn
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Abstract

We examine the Petviashvilli method for solving the equation $ \phi - \Delta \phi = |\phi|^{p-1} \phi$ on a bounded domain $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on $\mathbb{R}$ by Pelinovsky & Stepanyants, 2004. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.

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Derek Olson, Soumitra Shukla, Gideon Simpson, Daniel Spirn. 2014-11-15. Petviashvilli's Method for the Dirichlet Problem. https://arxiv.org/abs/1411.4153

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